Random, Stochastic, and Self-similar Equations

将使用广泛的数学方法来增加对发生在自相似、分形和无序介质中的随机过程的长期和短期行为的理解。证明了自相似Dirichlet形式、扩散和随机游动在一大类分形上的存在唯一性，其中包括表现为群的极限空间的无限分支分形。高斯型和非高斯型热核估计以及格林函数估计将被研究在无序系统上，如自相似和随机分形。此外，还将开发概率工具来研究具有局部Dirichlet形式的无序空间的非对易分析和广义微分几何。此外，该项目将有助于遍历理论的乘积不一定独立的矩阵及其与过程的局部性质的关系。我们将得到随机微分方程解的Lyapunov指数的渐近公式和随机微分方程解的Lyapunov指数的估计，并与随机微分方程谱问题和波在分形和其他无序介质中的传播有关。该项目有助于无序介质中过程的研究，在物理、化学、生物科学和工程中有许多应用。渗流簇中的扩散过程、分形物体的振动、带有随机障碍物的通道中的信号传播、分形天线中的电磁波、海洋学中的Rossby波、金融市场模型、神经结构只是这些过程的许多例子中的一小部分。因此，该项目有助于数学、物理、生物科学和工程学的融合。该项目将教育和研究与本科生融为一体。该项目的更广泛影响包括对科学和工程人力资源开发的贡献，扩大代表性不足群体的参与，以及加强研究和教育的基础设施。

A wide array of mathematical methods will be used to increase the understanding of the long and short term behavior of random processes occurring in self-similar, fractal and disordered media. The existence and uniqueness of self-similar Dirichlet forms, diffusions, and random walks will be proved on a wide class of fractals, including infinitely ramified fractals appearing as limit spaces of groups. Gaussian and non-Gaussian heat kernel estimates and Green's function estimates will be studied on disordered systems, such as self-similar and random fractals. Furthermore, probabilistic tools will be developed to study non-commutative analysis on and generalized differential geometry of disordered spaces that carry a local Dirichlet form. In addition, the project will contribute to the ergodic theory of products of not necessarily independent matrices and their relation to local properties of processes on fractals. Asymptotic formulas for Lyapunov exponents of differential and difference equations with small random perturbations, and estimates of the Lyapunov exponents of stochastic differential equations will be obtained, and related to the spectral problems for stochastic differential equations and wave propagation in fractal and other disordered media.The project contributes to the study of processes in disordered media (fractals), which have many applications in physics, chemistry, biological sciences and engineering. Diffusion processes in percolation clusters, vibrations of fractal objects, signal propagation in channels with random obstacles, electro-magnetic waves in fractal antennae, Rossby waves in oceanography, models of financial markets, neural structures are just a few of many examples of such processes. Thus the project contributes to the integration of mathematics, physics, biological sciences and engineering. The project integrates education and research with undergraduate students. The broader impacts of the project include contributions to the development of human resources in science and engineering, expanding participation of underrepresented groups, and enhancing infrastructure for research and education.